Some Characterizations of Sturmian Words in Terms of the Lexicographic Order

We investigate some repetition problems for a very special class $\mathcal{S}$ of strings called the standard Sturmian words, which have very compact representations in terms of sequences of integers. Usually the size of this word is exponential with respect to the size of its integer sequence, hence we are dealing with repetition problems in compressed strings. An explicit formula is given for the number $\rho(w)$ of runs in a standard word $w$. We show that $\rho(w)/|w|\le 4/5$ for each $w\in S$, and there is an infinite sequence of strictly growing words $w_k\in {\mathcal{S}}$ such that $\lim_{k\rightarrow \infty} \frac{\rho(w_k)}{|w_k|} = \frac{4}{5}$. Moreover, we show how to compute the number of runs in a standard Sturmian word in linear time with respect to the size of its compressed representation.

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A geometrical approach of palindromic factors of standard billiard words

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.410 ◽

2007 ◽

Vol Vol. 9 no. 2 ◽

Author(s):

Jean-Pierre Borel

Keyword(s):

Geometrical Approach ◽

International Audience ◽

Sturmian Words

International audience Many results are already known, concerning the palindromic factors and the palindomic prefixes of Standard billiard words, i.e., Sturmian words and billiard words in any dimension, starting at the origin. We give new geometrical proofs of these results, especially for the existence in any dimension of Standard billiard words with arbitrary long palindromic prefixes.

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On a Positivity Conjecture in the Character Table of $S_n$

The Electronic Journal of Combinatorics ◽

10.37236/8186 ◽

2019 ◽

Vol 26 (1) ◽

Author(s):

Sheila Sundaram

Keyword(s):

Lexicographic Order ◽

Character Table ◽

Power Sums ◽

Special Cases ◽

Schur Positive

In previous work of this author it was conjectured that the sum of power sums $p_\lambda,$ for partitions $\lambda$ ranging over an interval $[(1^n), \mu]$ in reverse lexicographic order, is Schur-positive. Here we investigate this conjecture and establish its truth in the following special cases: for $\mu\in [(n-4,1^4), (n)]$ or $\mu\in [(1^n), (3,1^{n-3})], $ or $\mu=(3, 2^k, 1^r)$ when $k\geq 1$ and $0\leq r\leq 2.$ Many new Schur positivity questions are presented.

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Most Probably Intersecting Hypergraphs

The Electronic Journal of Combinatorics ◽

10.37236/4784 ◽

2015 ◽

Vol 22 (1) ◽

Cited By ~ 1

Author(s):

Shagnik Das ◽

Benny Sudakov

Keyword(s):

Initial Segment ◽

Lexicographic Order ◽

Structural Characterisation ◽

Intersecting Families ◽

Intersecting Hypergraphs

The celebrated Erdős-Ko-Rado theorem shows that for $n \ge 2k$ the largest intersecting $k$-uniform set family on $[n]$ has size $\binom{n-1}{k-1}$. It is natural to ask how far from intersecting larger set families must be. Katona, Katona and Katona introduced the notion of most probably intersecting families, which maximise the probability of random subfamilies being intersecting.We consider the most probably intersecting problem for $k$-uniform set families. We provide a rough structural characterisation of the most probably intersecting families and, for families of particular sizes, show that the initial segment of the lexicographic order is optimal.

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